Exercise 26, § 7

a) Let $E$ be a Hausdorff, locally convex space and let $A$ be a cap of a convex set $C$ in $E$. If $s \in C$ is a point not belonging to $A$ and if $B$ is the cone with vertex $s$ generated by $A$ show that the closure of $B \cap (C \cap \complement A)$ is a cap in $B$. *b) Suppose that $E$ is finite dimensional. Show that every cap $A$ of a closed convex set $C$ in $E$ can be obtained in the following manner : consider a facet $F$ of $C$ (II, p. 87, exerc. 3) and a hyperplane $H$ in the affine linear variety $V$ generated by $F$, such that $F$ is entirely on one side of $H$, take as $A$ the set of points of $F$ contained between $H$ and a hyperplane $H'$ of $F$ parallel to $H$ (use a) and prop. 4 of II, p. 38). Every extremal point of a facet of $C$ is an extremal point of $C$. c) Give an example of a compact convex set $C$ in a Hausdorff locally convex space $E$ and of a cap $A$ of $C$ such that $A$ and $C \cap \complement A$ each generate $E$ and that $A$ and $C \cap \complement A$ cannot be separated by a closed hyperplane of $E$ (cf.* II, p. 78, exerc. 11).